1. **State the problem:** We need to find the expression for the perimeter of a rectangle where the width is $k + 3$ cm and the length is $k - 9$ cm. 2. **Recall the formula for perimeter of a rectangle:** $$P = 2(\text{length} + \text{width})$$ 3. **Substitute the given expressions:** $$P = 2((k - 9) + (k + 3))$$ 4. **Simplify inside the parentheses:** $$P = 2(k - 9 + k + 3)$$ $$P = 2(2k - 6)$$ 5. **Distribute the 2:** $$P = 2 \times 2k - 2 \times 6$$ $$P = 4k - 12$$ 6. **Final answer:** The expression representing the perimeter is $4k - 12$. This matches the option $-12 + 4k$ (which is the same expression rearranged).